Complement can also be used as a verb. It means to complete or enhance something by providing something additional.

Example: The tasty dinner was complemented by a splendid dessert.

Complementary

Complementary is an adjective. It means completing something else or making it better.

Example: The company and its partners have a complementary goal to serve their clients better.

Conclusion

These are homophones (with the same pronunciation) and their usages can be confusing. If you’re saying nice things about people, it is compliment/complimentary. If you’re talking about completing or enhancing something – it is complement/complementary.